Thermal effects in field electron emission from idealized arrangements of independent and interacting micro-protrusions

Abstract

Modelling studies of thermo-field electron emission (TFE) from protrusions at a cathode surface usually use simulations in 2D axial symmetry. Indeed, time-dependent simulations in 3D are very demanding in computation time. Often, 3D simulations have been restricted to stationary pure field electron emission to account for the drastic current decrease caused by electric field screening when the emitters are close. Little interest has therefore been granted to the heat exchanges occurring between nearby emitters. Although the temperature is a second-order parameter in TFE compared to the electric field, thermal effects become non-negligible in high current density regimes, where self-heating is well established. The present study focuses on the thermal effects occurring during the TFE from micro-protrusions. Our model considers a DC voltage but solves in time the temperature evolution coupling the heat equation and the current continuity equation. The protrusions are modelled as hemiellipsoids with 2D axial symmetry. Emission enhancement due to the increase of the temperature in the thermo-field regime compared to the pure field regime is detailed as a test case for isolated protrusions. Then, full 3D simulations are used to investigate the thermal coupling between multiple neighbouring protrusions via their outwards heat fluxes inside the cathode. The results show a higher current increase due to thermal coupling for dome-like protrusions with a low field enhancement factor. The current increases up to 13% of the total current for aspect ratios of 1, but this value is reached for an extreme applied electric field, hardly reachable in experiments. For sharper protrusions with higher field enhancement, the interaction range through the cathode being shorter, the thermal coupling is suppressed by electrostatic screening. Nevertheless, in arrangements of densely distributed field emitter, when the screening is compensated by a higher voltage, our model predicts the possibility of a moderate but noticeable thermal coupling even for sharp protrusions: a parametric study indicates up to 14.5% of the emitted current being caused by a thermal coupling through the cathode bulk, for protrusions with an aspect ratio of 10 under a fixed applied electric field of 0.4 GV m−1 in DC mode.